Question

you are given the sample mean and the population standard deviation. use this information to construct the 90% and 95% confidence intervals for the population mean. interpret the results and compare the widths of the confidence intervals.
from a random sample of 37 months from january 2006 through december 2020, the mean number of tornadoes per month in the united states was about 101. assume the population standard deviation is 115.
the 90% confidence interval is (69.89, 132.11).
(round to two decimal places as needed.)
the 95% confidence interval is (□,□).
(round to two decimal places as needed.)

Explanation:

Step1: Calculate the margin of error for 95% confidence interval

The formula for the margin of error \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). For a 95% confidence interval, \(z_{\alpha/2}=1.96\), \(\sigma = 115\), \(n = 37\).
\(E=1.96\times\frac{115}{\sqrt{37}}\)
First, calculate \(\sqrt{37}\approx6.08\), then \(\frac{115}{6.08}\approx18.91\), and \(E = 1.96\times18.91\approx37.06\)
The sample mean \(\bar{x}=\frac{69.89 + 132.11}{2}=\frac{202}{2}=101\)
The 95% confidence interval is \(\bar{x}\pm E\), so \(101- 37.06=63.94\) and \(101 + 37.06=138.06\)

Answer:

\((63.94,138.06)\)